Link!

Remember in Back to the Future (I, II, and III) Doc Brown used a “flux capacitor” to travel through time?  Well, what if I told you that every capacitor is a flux capacitor?

Well that’s true.   As a simple example, say I took a capacitor and hooked it up to a battery, as below.  Then I show the capacitor as two metal plates with electric field between them, and a surface “S” parallel to the two plates.

As this crude analysis points out, the flux is nonzero so long as the voltage across the capacitor – any capacitor  – is nonzero.

Movie fans will point out that this isn’t magnetic flux, to which I have two rebuttals:

1.) the movie never states whether Doc Brown means electric or magnetic flux

2.) fine, you want a magnetic flux capacitor?  I give you, the mystical LC network – the transmission line stub, the antenna, the cavity resonator…etc.

Sorry Robert Zemeckis.

Below is a function I used to plot smith chart results in matlab.  I used basic plotting code to generate the chart itself, and added a simply plot function to add the impedance locus and constant VSWR circles.  Hope this is helpeful!  Please let me know if anyone finds a bug.  I will note that there is no safeguard in the function for s-parameters that are greater than 1, which if you are plotting the active S-parameters of a multiport device, are possible.  This isn’t a big deal when the S-parameter goes slightly above 1, but if it swings well above 1 you end up with a tiny smith chart and these erratic line segments…it’s a mess.  For passive applications I haven’t come across a bug yet.

smithchart.m

So I decided that, being able to compute all the data I need from S-parameters alone (as HFSS does), I exported the entire S-matrix to MATLAB and took a look at the impedance values I calculated.

HFSS gives the equation for finding the Z matrix as :

Z = sqrt(Zo*I)*inv(I-S)*(I+S)*sqrt(Zo*I)

where I=identity matrix, and S is the full S-parameter matrix

from this I plotted the exact same thing HFSS was giving for the impedances.  So I started plotting different chunks of the equation (e.g. inv(I-s) alone, (I+S) alone…) and found that the inv(I-S) term was spiking at the exact places my impedance was spiking, while the other terms were well-behaved.  Well, looks like that inversion is messing things up…indicating a singular, or more correctly a nearly-singular matrix (singular matrices have NO inverse defined).  Sure enough, using the function:

cond()

in MATLAB revealed huge condition numbers for the matrix (I-S) where the impedance spikes occurred, 2 orders of magnitude larger than the condition numbers calculated over the rest of the frequency sweep.  So I need to invert a matrix that is nearly-singular….now what??  I need single-value-decomposition (SVD) to take out those values that make the matrix singular and do a quasi-inversion, basically approximate the inversion.  MATLAB includes just such a function:

pinv() — where it’s use is  pinv(matrix, tolerance)

The tolerance can be set to an arbitrary value.  For my purposes, I can stand to lose some accuracy, so I chose a relatively large tolerance (around 0.02 or so) to take out those spikes but preserve the overal shape of the impedance waveforms.  Usually you will want to keep this very small to keep the inversion close to the true value.  But this solves my problem.

But it leads to another conclusion — HFSS, apparently, doesn’t test if its S-parameter matrices are nearly-singular (or at least I haven’t seen a way to have it do so).  Certainly it is using all sorts of inversion approximations in the solver the software uses (you can even do iterative matrix solutions in v11 of HFSS…), I don’t think it is likely that they have skimped on such checking on the data output front end.   Please post comments if you konw of any solutions in HFSS for this.

1 + 3 + 5 + 7… is an arithmetic progression because the difference between all terms is two.

Anyway, by summing this series up he figured out the day he would sleep for 24 hours, and declared that as the day he would die. He died on that very day – November 27th, 1754.